Open Access
2023 Probabilistic representations of fragmentation equations
Madalina Deaconu, Antoine Lejay
Author Affiliations +
Probab. Surveys 20: 226-290 (2023). DOI: 10.1214/23-PS14
Abstract

In this survey article, we present an overview of a large class of probabilistic representations of the fragmentation equation, and we develop and study the interconnections in between these representations. We focus on the stochastic process which represents the evolution of the mass of a typical particle which undergoes fragmentation in time. These probabilistic representations range from Markov chains to stochastic differential equations with jumps, and we aim at constructing how they are inter-related. In particular, we show how these representations can be used to develop easy to implement numerical methods.

References

1.

Ackleh, A. S., Lyons, R. and Saintier, N. (2021). A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models. ESAIM Math. Model. Numer. Anal. 55 2473–2501.  MR4330732Ackleh, A. S., Lyons, R. and Saintier, N. (2021). A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models. ESAIM Math. Model. Numer. Anal. 55 2473–2501.  MR4330732

2.

Ackleh, A. S. and Saintier, N. (2020). Well-posedness of a system of transport and diffusion equations in space of measures. J. Math. Anal. Appl. 492 124397, 28.  MR4126768Ackleh, A. S. and Saintier, N. (2020). Well-posedness of a system of transport and diffusion equations in space of measures. J. Math. Anal. Appl. 492 124397, 28.  MR4126768

3.

Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48.  MR1673235Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48.  MR1673235

4.

Azaïs, R. (2014). A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process. ESAIM Probab. Stat. 18 726–749.  MR3334012Azaïs, R. (2014). A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process. ESAIM Probab. Stat. 18 726–749.  MR3334012

5.

Azaïs, R. and Muller-Gueudin, A. (2016). Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes. Electron. J. Stat. 10 3648–3692.  MR3579198Azaïs, R. and Muller-Gueudin, A. (2016). Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes. Electron. J. Stat. 10 3648–3692.  MR3579198

6.

Bailleul, I. F. (2011). Sensitivity for the Smoluchowski equation. J. Phys. A 44 245004, 21.  MR2800857Bailleul, I. F. (2011). Sensitivity for the Smoluchowski equation. J. Phys. A 44 245004, 21.  MR2800857

7.

Bailleul, I. F., Man, P. L. W. and Kraft, M. (2010). A stochastic algorithm for parametric sensitivity in Smoluchowski’s coagulation equation. SIAM J. Numer. Anal. 48 1064–1086.  MR2679572Bailleul, I. F., Man, P. L. W. and Kraft, M. (2010). A stochastic algorithm for parametric sensitivity in Smoluchowski’s coagulation equation. SIAM J. Numer. Anal. 48 1064–1086.  MR2679572

8.

Banasiak, J. (2006). Shattering and non-uniqueness in fragmentation models—an analytic approach. Phys. D 222 63–72.  MR2265768Banasiak, J. (2006). Shattering and non-uniqueness in fragmentation models—an analytic approach. Phys. D 222 63–72.  MR2265768

9.

Banasiak, J. and Arlotti, L. (2006). Perturbations of Positive Semigroups with Applications. Springer Monographs in Mathematics. Springer London, Ltd., London.  MR2178970Banasiak, J. and Arlotti, L. (2006). Perturbations of Positive Semigroups with Applications. Springer Monographs in Mathematics. Springer London, Ltd., London.  MR2178970

10.

Bass, R. F. (2004). Stochastic differential equations with jumps. Probab. Surv. 1 1–19.  MR2095564Bass, R. F. (2004). Stochastic differential equations with jumps. Probab. Surv. 1 1–19.  MR2095564

11.

Berestycki, J., Bertoin, J., Haas, B. and Miermont, G. (2010). Quelques aspects fractals des fragmentations aléatoires. In Quelques Interactions Entre Analyse, Probabilités et Fractals. Panor. Synthèses 32 191–243. Soc. Math. France, Paris.  MR2932439Berestycki, J., Bertoin, J., Haas, B. and Miermont, G. (2010). Quelques aspects fractals des fragmentations aléatoires. In Quelques Interactions Entre Analyse, Probabilités et Fractals. Panor. Synthèses 32 191–243. Soc. Math. France, Paris.  MR2932439

12.

Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.  MR2253162Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.  MR2253162

13.

Bertoin, J. (2006). Different aspects of a random fragmentation model. Stochastic Process. Appl. 116 345–369.  MR2199553Bertoin, J. (2006). Different aspects of a random fragmentation model. Stochastic Process. Appl. 116 345–369.  MR2199553

14.

Bertoin, J. (2017). Markovian growth-fragmentation processes. Bernoulli 23 1082–1101.  MR3606760Bertoin, J. (2017). Markovian growth-fragmentation processes. Bernoulli 23 1082–1101.  MR3606760

15.

Bertoin, J. (2019). On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors. J. Funct. Anal. 277 108270, 29.  MR4013826Bertoin, J. (2019). On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors. J. Funct. Anal. 277 108270, 29.  MR4013826

16.

Beznea, L., Deaconu, M. and Lupaşcu, O. (2015). Branching processes for the fragmentation equation. Stochastic Process. Appl. 125 1861–1885.  MR3315615Beznea, L., Deaconu, M. and Lupaşcu, O. (2015). Branching processes for the fragmentation equation. Stochastic Process. Appl. 125 1861–1885.  MR3315615

17.

Beznea, L., Deaconu, M. and Lupaşcu, O. (2016). Stochastic equation of fragmentation and branching processes related to avalanches. J. Stat. Phys. 162 824–841.  MR3456978Beznea, L., Deaconu, M. and Lupaşcu, O. (2016). Stochastic equation of fragmentation and branching processes related to avalanches. J. Stat. Phys. 162 824–841.  MR3456978

18.

Beznea, L., Deaconu, M. and Lupaşcu-Stamate, O. (2019). Numerical approach for stochastic differential equations of fragmentation; application to avalanches. Math. Comput. Simulation 160 111–125.  MR3926782Beznea, L., Deaconu, M. and Lupaşcu-Stamate, O. (2019). Numerical approach for stochastic differential equations of fragmentation; application to avalanches. Math. Comput. Simulation 160 111–125.  MR3926782

19.

Bogachev, V. I. (2007). Measure Theory. Vol. I, II. Springer, Berlin.  MR2267655Bogachev, V. I. (2007). Measure Theory. Vol. I, II. Springer, Berlin.  MR2267655

20.

Bouguet, F. (2018). A probabilistic look at conservative growth-fragmentation equations. In Séminaire de Probabilités XLIX. Lecture Notes in Math. 2215 57–74. Springer, Cham.  MR3837101Bouguet, F. (2018). A probabilistic look at conservative growth-fragmentation equations. In Séminaire de Probabilités XLIX. Lecture Notes in Math. 2215 57–74. Springer, Cham.  MR3837101

21.

Bouteiller, C. L., Naaim-Bouvet, F., Mathys, N. and Lavé, J. (2011). A new framework for modeling sediment fining during transport with fragmentation and abrasion. Journal of Geophysical Research 116.  10.1029/2010jf001926Bouteiller, C. L., Naaim-Bouvet, F., Mathys, N. and Lavé, J. (2011). A new framework for modeling sediment fining during transport with fragmentation and abrasion. Journal of Geophysical Research 116.  10.1029/2010jf001926

22.

Cañizo, J. A., Gabriel, P. and Yoldaş, H. (2021). Spectral gap for the growth-fragmentation equation via Harris’s theorem. SIAM J. Math. Anal. 53 5185–5214.  MR4312814Cañizo, J. A., Gabriel, P. and Yoldaş, H. (2021). Spectral gap for the growth-fragmentation equation via Harris’s theorem. SIAM J. Math. Anal. 53 5185–5214.  MR4312814

23.

Cavalli, B. (2021). A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates. ESAIM Probab. Stat. 25 258–285.  MR4268042Cavalli, B. (2021). A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates. ESAIM Probab. Stat. 25 258–285.  MR4268042

24.

Cavalli, B. (2021). A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates. ESAIM Probab. Stat. 25 258–285.  MR4268042Cavalli, B. (2021). A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates. ESAIM Probab. Stat. 25 258–285.  MR4268042

25.

Çinlar, E. and Jacod, J. (1981). Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981). Progr. Prob. Statist. 1 159–242. Birkhäuser, Boston, Mass.  MR0647786Çinlar, E. and Jacod, J. (1981). Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981). Progr. Prob. Statist. 1 159–242. Birkhäuser, Boston, Mass.  MR0647786

26.

Çinlar, E., Jacod, J., Protter, P. and Sharpe, M. J. (1980). Semimartingales and Markov processes. Z. Wahrsch. Verw. Gebiete 54 161–219.  MR0597337Çinlar, E., Jacod, J., Protter, P. and Sharpe, M. J. (1980). Semimartingales and Markov processes. Z. Wahrsch. Verw. Gebiete 54 161–219.  MR0597337

27.

Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I, 2nd ed. Probability and Its Applications (New York). Springer, New York. Elementary theory and methods.  MR1950431Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I, 2nd ed. Probability and Its Applications (New York). Springer, New York. Elementary theory and methods.  MR1950431

28.

Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II, 2nd ed. Probability and Its Applications (New York). Springer, New York. General theory and structure.  MR2371524Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II, 2nd ed. Probability and Its Applications (New York). Springer, New York. General theory and structure.  MR2371524

29.

Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B 46 353–388. With discussion.  MR0790622Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B 46 353–388. With discussion.  MR0790622

30.

Devroye, L. (1986). Nonuniform Random Variate Generation. Springer, New York.  MR0836973Devroye, L. (1986). Nonuniform Random Variate Generation. Springer, New York.  MR0836973

31.

Diemer, R. B. and Olson, J. H. (2002). A moment methodology for coagulation and breakage problems: Part 3—generalized daughter distribution functions. Chemical Engineering Science 57 4187–4198.  10.1016/S0009-2509(02)00366-4Diemer, R. B. and Olson, J. H. (2002). A moment methodology for coagulation and breakage problems: Part 3—generalized daughter distribution functions. Chemical Engineering Science 57 4187–4198.  10.1016/S0009-2509(02)00366-4

32.

Doumic, M., Escobedo, M. and Tournus, M. (2018). Estimating the division rate and kernel in the fragmentation equation. Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 1847–1884.  MR3906858Doumic, M., Escobedo, M. and Tournus, M. (2018). Estimating the division rate and kernel in the fragmentation equation. Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 1847–1884.  MR3906858

33.

Dubovski˘ı, P. B. and Stewart, I. W. (1996). Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Methods Appl. Sci. 19 571–591.  MR1385155Dubovski˘ı, P. B. and Stewart, I. W. (1996). Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Methods Appl. Sci. 19 571–591.  MR1385155

34.

Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge. Revised reprint of the 1989 original.  MR1932358Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge. Revised reprint of the 1989 original.  MR1932358

35.

El Karoui, N. and Lepeltier, J.-P. (1977). Représentation des processus ponctuels multivariés à l’aide d’un processus de Poisson. Z. Wahrsch. Verw. Gebiete 39 111–133.  MR0448546El Karoui, N. and Lepeltier, J.-P. (1977). Représentation des processus ponctuels multivariés à l’aide d’un processus de Poisson. Z. Wahrsch. Verw. Gebiete 39 111–133.  MR0448546

36.

Engel, K.-J. and Nagel, R. (2000). One-parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194. Springer, New York. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.  MR1721989Engel, K.-J. and Nagel, R. (2000). One-parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194. Springer, New York. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.  MR1721989

37.

Erickson, R. V. (1974). Paths of random evolutions. Z. Wahrsch. Verw. Gebiete 29 309–321.  MR0418252Erickson, R. V. (1974). Paths of random evolutions. Z. Wahrsch. Verw. Gebiete 29 309–321.  MR0418252

38.

Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York. Characterization and convergence.  MR0838085Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York. Characterization and convergence.  MR0838085

39.

Feinberg, E. A., Mandava, M. and Shiryaev, A. N. (2014). On solutions of Kolmogorov’s equations for nonhomogeneous jump Markov processes. J. Math. Anal. Appl. 411 261–270.  MR3118483Feinberg, E. A., Mandava, M. and Shiryaev, A. N. (2014). On solutions of Kolmogorov’s equations for nonhomogeneous jump Markov processes. J. Math. Anal. Appl. 411 261–270.  MR3118483

40.

Feller, W. (1940). On the integro-differential equations of purely discontinuous Markoff processes. Trans. Amer. Math. Soc. 48 488–515.  MR0002697Feller, W. (1940). On the integro-differential equations of purely discontinuous Markoff processes. Trans. Amer. Math. Soc. 48 488–515.  MR0002697

41.

Feller, W. (1945). Erratum to “On the integro-differential equations of purely discontinuous Markoff processes”. Trans. Amer. Math. Soc. 58 474–474.  10.1090/S0002-9947-45-99962-4Feller, W. (1945). Erratum to “On the integro-differential equations of purely discontinuous Markoff processes”. Trans. Amer. Math. Soc. 58 474–474.  10.1090/S0002-9947-45-99962-4

42.

Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.  MR0270403Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.  MR0270403

43.

Filippov, A. F. (1961). Über das Verteilungsgesetz der Grössen der Teilchen bei Zerstückelung. Teor. Verojatnost. i Primenen. 6 299–318.  MR0140159Filippov, A. F. (1961). Über das Verteilungsgesetz der Grössen der Teilchen bei Zerstückelung. Teor. Verojatnost. i Primenen. 6 299–318.  MR0140159

44.

Fournier, N. and Giet, J.-S. (2003). On small particles in coagulation-fragmentation equations. J. Stat. Phys. 111 1299–1329.  MR1975930Fournier, N. and Giet, J.-S. (2003). On small particles in coagulation-fragmentation equations. J. Stat. Phys. 111 1299–1329.  MR1975930

45.

Grandell, J. (1977). Point processes and random measures. Advances in Appl. Probability 9 502–526.  MR0478331Grandell, J. (1977). Point processes and random measures. Advances in Appl. Probability 9 502–526.  MR0478331

46.

Grandell, J. (1977). Point processes and random measures. Advances in Appl. Probability 9 502–526.  MR0478331Grandell, J. (1977). Point processes and random measures. Advances in Appl. Probability 9 502–526.  MR0478331

47.

Haas, B. (2003). Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 245–277.  MR1989629Haas, B. (2003). Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 245–277.  MR1989629

48.

Haas, B. (2004). Appearance of dust in fragmentations. Commun. Math. Sci. 2 65–73.  MR2119874Haas, B. (2004). Appearance of dust in fragmentations. Commun. Math. Sci. 2 65–73.  MR2119874

49.

Haas, B. (2004). Regularity of formation of dust in self-similar fragmentations. Ann. Inst. Henri Poincaré Probab. Stat. 40 411–438.  MR2070333Haas, B. (2004). Regularity of formation of dust in self-similar fragmentations. Ann. Inst. Henri Poincaré Probab. Stat. 40 411–438.  MR2070333

50.

Hoang, V. H., Pham Ngoc, T. M., Rivoirard, V. and Tran, V. C. (2022). Nonparametric estimation of the fragmentation kernel based on a partial differential equation stationary distribution approximation. Scand. J. Stat. 49 4–43.  MR4391046Hoang, V. H., Pham Ngoc, T. M., Rivoirard, V. and Tran, V. C. (2022). Nonparametric estimation of the fragmentation kernel based on a partial differential equation stationary distribution approximation. Scand. J. Stat. 49 4–43.  MR4391046

51.

Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. North-Holland, Amsterdam-New York; Kodansha, Ltd., Tokyo.  MR0637061Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. North-Holland, Amsterdam-New York; Kodansha, Ltd., Tokyo.  MR0637061

52.

Ionescu Tulcea, C. T. (1949). Mesures dan les espaces produits. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 7 208–211 (1950).  MR0036288Ionescu Tulcea, C. T. (1949). Mesures dan les espaces produits. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 7 208–211 (1950).  MR0036288

53.

Jacod, J. (1974/75). Multivariate point processes: Predictable projection, Radon-Nikodým derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235–253.  MR0380978Jacod, J. (1974/75). Multivariate point processes: Predictable projection, Radon-Nikodým derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235–253.  MR0380978

54.

Jacod, J. (1977). Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales. In Séminaire de Probabilités, XI (Univ. Strasbourg, Strasbourg, 1975/1976). Lecture Notes in Math., Vol. 581 390–410. Springer, Berlin.  MR0458578Jacod, J. (1977). Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales. In Séminaire de Probabilités, XI (Univ. Strasbourg, Strasbourg, 1975/1976). Lecture Notes in Math., Vol. 581 390–410. Springer, Berlin.  MR0458578

55.

Jacod, J. and Skorokhod, A. V. (1996). Jumping Markov processes. Ann. Inst. Henri Poincaré Probab. Stat. 32 11–67.  MR1373726Jacod, J. and Skorokhod, A. V. (1996). Jumping Markov processes. Ann. Inst. Henri Poincaré Probab. Stat. 32 11–67.  MR1373726

56.

Jeon, I. (1998). Existence of gelling solutions for coagulation-fragmentation equations. Comm. Math. Phys. 194 541–567.  MR1631473Jeon, I. (1998). Existence of gelling solutions for coagulation-fragmentation equations. Comm. Math. Phys. 194 541–567.  MR1631473

57.

Jourdain, B. (2004). Uniqueness via probabilistic interpretation for the discrete coagulation fragmentation equation. Commun. Math. Sci. 2 75–83.  MR2119875Jourdain, B. (2004). Uniqueness via probabilistic interpretation for the discrete coagulation fragmentation equation. Commun. Math. Sci. 2 75–83.  MR2119875

58.

Kallenberg, O. (2021). Foundations of Modern Probability. Probability Theory and Stochastic Modelling 99. Springer, Cham. Third edition [of 1464694].  MR4226142Kallenberg, O. (2021). Foundations of Modern Probability. Probability Theory and Stochastic Modelling 99. Springer, Cham. Third edition [of 1464694].  MR4226142

59.

Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. The Clarendon Press, Oxford University Press, New York. Oxford Science Publications.  MR1207584Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. The Clarendon Press, Oxford University Press, New York. Oxford Science Publications.  MR1207584

60.

Kolmogoroff, A. N. (1941). Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. C. R. (Doklady) Acad. Sci. URSS (N. S.) 31 99–101.  MR0004415Kolmogoroff, A. N. (1941). Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. C. R. (Doklady) Acad. Sci. URSS (N. S.) 31 99–101.  MR0004415

61.

Kolokoltsov, V. N. (2006). Kinetic equations for the pure jump models of k-nary interacting particle systems. Markov Process. Related Fields 12 95–138.  MR2223422Kolokoltsov, V. N. (2006). Kinetic equations for the pure jump models of k-nary interacting particle systems. Markov Process. Related Fields 12 95–138.  MR2223422

62.

Kurtz, T. G. (2014). Weak and strong solutions of general stochastic models. Electron. Commun. Probab. 19 no. 58, 16.  MR3254737Kurtz, T. G. (2014). Weak and strong solutions of general stochastic models. Electron. Commun. Probab. 19 no. 58, 16.  MR3254737

63.

Lamb, W. (2004). Existence and uniqueness results for the continuous coagulation and fragmentation equation. Math. Methods Appl. Sci. 27 703–721.  MR2070223Lamb, W. (2004). Existence and uniqueness results for the continuous coagulation and fragmentation equation. Math. Methods Appl. Sci. 27 703–721.  MR2070223

64.

Laurençot, P. (2000). On a class of continuous coagulation-fragmentation equations. J. Differential Equations 167 245–274.  MR1793195Laurençot, P. (2000). On a class of continuous coagulation-fragmentation equations. J. Differential Equations 167 245–274.  MR1793195

65.

Lemaire, V., Thieullen, M. and Thomas, N. (2020). Thinning and multilevel Monte Carlo methods for piecewise deterministic (Markov) processes with an application to a stochastic Morris-Lecar model. Adv. in Appl. Probab. 52 138–172.  MR4092810Lemaire, V., Thieullen, M. and Thomas, N. (2020). Thinning and multilevel Monte Carlo methods for piecewise deterministic (Markov) processes with an application to a stochastic Morris-Lecar model. Adv. in Appl. Probab. 52 138–172.  MR4092810

66.

Lewis, P. A. W. and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. Quart. 26 403–413.  MR0546120Lewis, P. A. W. and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. Quart. 26 403–413.  MR0546120

67.

Man, P. L. W., Norris, J. R., Bailleul, I. F. and Kraft, M. (2010). Coupling algorithms for calculating sensitivities of Smoluchowski’s coagulation equation. SIAM J. Sci. Comput. 32 635–655.  MR2609334Man, P. L. W., Norris, J. R., Bailleul, I. F. and Kraft, M. (2010). Coupling algorithms for calculating sensitivities of Smoluchowski’s coagulation equation. SIAM J. Sci. Comput. 32 635–655.  MR2609334

68.

Martinez-Bazán, C., Rodriguez-Rodriguez, J., Deane, G. B., Montañes, J. L. and Lasheras, J. C. (2010). Considerations on bubble fragmentation models. Journal of Fluid Mechanics 661 159–177.  10.1017/s0022112010003186Martinez-Bazán, C., Rodriguez-Rodriguez, J., Deane, G. B., Montañes, J. L. and Lasheras, J. C. (2010). Considerations on bubble fragmentation models. Journal of Fluid Mechanics 661 159–177.  10.1017/s0022112010003186

69.

McGrady, E. D. and Ziff, R. M. (1987). “Shattering” transition in fragmentation. Phys. Rev. Lett. 58 892–895.  MR0927489McGrady, E. D. and Ziff, R. M. (1987). “Shattering” transition in fragmentation. Phys. Rev. Lett. 58 892–895.  MR0927489

70.

McLaughlin, D. J., Lamb, W. and McBride, A. C. (1997). A semigroup approach to fragmentation models. SIAM J. Math. Anal. 28 1158–1172.  MR1466674McLaughlin, D. J., Lamb, W. and McBride, A. C. (1997). A semigroup approach to fragmentation models. SIAM J. Math. Anal. 28 1158–1172.  MR1466674

71.

Melzak, Z. A. (1957). A scalar transport equation. Trans. Amer. Math. Soc. 85 547–560.  MR0087880Melzak, Z. A. (1957). A scalar transport equation. Trans. Amer. Math. Soc. 85 547–560.  MR0087880

72.

Ogata, Y. (1981). On Lewis’ simulation method for point processes. IEEE transactions on information theory 27 23–31. .Ogata, Y. (1981). On Lewis’ simulation method for point processes. IEEE transactions on information theory 27 23–31. .

73.

Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge. Itô calculus, Reprint of the second (1994) edition.  MR1780932Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge. Itô calculus, Reprint of the second (1994) edition.  MR1780932

74.

Stewart, I. W. (1989). A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels. Math. Methods Appl. Sci. 11 627–648.  MR1011810Stewart, I. W. (1989). A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels. Math. Methods Appl. Sci. 11 627–648.  MR1011810

75.

Tyran-Kamińska, M. (2009). Substochastic semigroups and densities of piecewise deterministic Markov processes. J. Math. Anal. Appl. 357 385–402.  MR2557653Tyran-Kamińska, M. (2009). Substochastic semigroups and densities of piecewise deterministic Markov processes. J. Math. Anal. Appl. 357 385–402.  MR2557653

76.

Varadarajan, V. S. (1958). Weak convergence of measures on separable metric spaces. Sankhy¯a 19 15–22.  MR0094838Varadarajan, V. S. (1958). Weak convergence of measures on separable metric spaces. Sankhy¯a 19 15–22.  MR0094838

77.

Wagner, W. (2005). Explosion phenomena in stochastic coagulation-fragmentation models. Ann. Appl. Probab. 15 2081–2112.  MR2152254Wagner, W. (2005). Explosion phenomena in stochastic coagulation-fragmentation models. Ann. Appl. Probab. 15 2081–2112.  MR2152254

78.

Wagner, W. (2010). Random and deterministic fragmentation models. Monte Carlo Methods Appl. 16 399–420.  MR2747823Wagner, W. (2010). Random and deterministic fragmentation models. Monte Carlo Methods Appl. 16 399–420.  MR2747823

79.

Wieczorek, R. (2015). A stochastic particles model of fragmentation process with shattering. Electron. J. Probab. 20 no. 86, 17.  MR3391869Wieczorek, R. (2015). A stochastic particles model of fragmentation process with shattering. Electron. J. Probab. 20 no. 86, 17.  MR3391869

80.

Zhang, Y. (2018). On the nonexplosion and explosion for nonhomogeneous Markov pure jump processes. J. Theoret. Probab. 31 1322–1355.  MR3842155Zhang, Y. (2018). On the nonexplosion and explosion for nonhomogeneous Markov pure jump processes. J. Theoret. Probab. 31 1322–1355.  MR3842155
Madalina Deaconu and Antoine Lejay "Probabilistic representations of fragmentation equations," Probability Surveys 20(none), 226-290, (2023). https://doi.org/10.1214/23-PS14
Received: 1 December 2021; Published: 2023
Vol.20 • 2023
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